3.1.63 \(\int \frac {1}{(a+b (F^{g (e+f x)})^n)^3 (c+d x)^2} \, dx\) [63]

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {1}{\left (a+b \left (F^{e g+f g x}\right )^n\right )^3 (c+d x)^2},x\right ) \]

[Out]

Unintegrable(1/(a+b*(F^(f*g*x+e*g))^n)^3/(d*x+c)^2,x)

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((a + b*(F^(g*(e + f*x)))^n)^3*(c + d*x)^2),x]

[Out]

Defer[Int][1/((a + b*(F^(e*g + f*g*x))^n)^3*(c + d*x)^2), x]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2} \, dx &=\int \frac {1}{\left (a+b \left (F^{e g+f g x}\right )^n\right )^3 (c+d x)^2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3 (c+d x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((a + b*(F^(g*(e + f*x)))^n)^3*(c + d*x)^2),x]

[Out]

Integrate[1/((a + b*(F^(g*(e + f*x)))^n)^3*(c + d*x)^2), x]

________________________________________________________________________________________

Maple [A]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )^{3} \left (d x +c \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*(F^(g*(f*x+e)))^n)^3/(d*x+c)^2,x)

[Out]

int(1/(a+b*(F^(g*(f*x+e)))^n)^3/(d*x+c)^2,x)

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*(F^(g*(f*x+e)))^n)^3/(d*x+c)^2,x, algorithm="maxima")

[Out]

1/2*(3*a*d*f*g*n*x*log(F) + 3*a*c*f*g*n*log(F) + 2*(F^(g*n*e)*b*d*f*g*n*x*log(F) + F^(g*n*e)*b*c*f*g*n*log(F)
+ F^(g*n*e)*b*d)*F^(f*g*n*x) + 2*a*d)/(a^4*d^3*f^2*g^2*n^2*x^3*log(F)^2 + 3*a^4*c*d^2*f^2*g^2*n^2*x^2*log(F)^2
+ 3*a^4*c^2*d*f^2*g^2*n^2*x*log(F)^2 + a^4*c^3*f^2*g^2*n^2*log(F)^2 + (F^(2*g*n*e)*a^2*b^2*d^3*f^2*g^2*n^2*x^
3*log(F)^2 + 3*F^(2*g*n*e)*a^2*b^2*c*d^2*f^2*g^2*n^2*x^2*log(F)^2 + 3*F^(2*g*n*e)*a^2*b^2*c^2*d*f^2*g^2*n^2*x*
log(F)^2 + F^(2*g*n*e)*a^2*b^2*c^3*f^2*g^2*n^2*log(F)^2)*F^(2*f*g*n*x) + 2*(F^(g*n*e)*a^3*b*d^3*f^2*g^2*n^2*x^
3*log(F)^2 + 3*F^(g*n*e)*a^3*b*c*d^2*f^2*g^2*n^2*x^2*log(F)^2 + 3*F^(g*n*e)*a^3*b*c^2*d*f^2*g^2*n^2*x*log(F)^2
+ F^(g*n*e)*a^3*b*c^3*f^2*g^2*n^2*log(F)^2)*F^(f*g*n*x)) + integrate((d^2*f^2*g^2*n^2*x^2*log(F)^2 + c^2*f^2*
g^2*n^2*log(F)^2 + 3*c*d*f*g*n*log(F) + 3*d^2 + (2*c*d*f^2*g^2*n^2*log(F)^2 + 3*d^2*f*g*n*log(F))*x)/(a^3*d^4*
f^2*g^2*n^2*x^4*log(F)^2 + 4*a^3*c*d^3*f^2*g^2*n^2*x^3*log(F)^2 + 6*a^3*c^2*d^2*f^2*g^2*n^2*x^2*log(F)^2 + 4*a
^3*c^3*d*f^2*g^2*n^2*x*log(F)^2 + a^3*c^4*f^2*g^2*n^2*log(F)^2 + (F^(g*n*e)*a^2*b*d^4*f^2*g^2*n^2*x^4*log(F)^2
+ 4*F^(g*n*e)*a^2*b*c*d^3*f^2*g^2*n^2*x^3*log(F)^2 + 6*F^(g*n*e)*a^2*b*c^2*d^2*f^2*g^2*n^2*x^2*log(F)^2 + 4*F
^(g*n*e)*a^2*b*c^3*d*f^2*g^2*n^2*x*log(F)^2 + F^(g*n*e)*a^2*b*c^4*f^2*g^2*n^2*log(F)^2)*F^(f*g*n*x)), x)

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*(F^(g*(f*x+e)))^n)^3/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral(1/(a^3*d^2*x^2 + 2*a^3*c*d*x + a^3*c^2 + (b^3*d^2*x^2 + 2*b^3*c*d*x + b^3*c^2)*(F^(f*g*x + g*e))^(3*n
) + 3*(a*b^2*d^2*x^2 + 2*a*b^2*c*d*x + a*b^2*c^2)*(F^(f*g*x + g*e))^(2*n) + 3*(a^2*b*d^2*x^2 + 2*a^2*b*c*d*x +
a^2*b*c^2)*(F^(f*g*x + g*e))^n), x)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*(F**(g*(f*x+e)))**n)**3/(d*x+c)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*(F^(g*(f*x+e)))^n)^3/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate(1/(((F^((f*x + e)*g))^n*b + a)^3*(d*x + c)^2), x)

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^3\,{\left (c+d\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*(F^(g*(e + f*x)))^n)^3*(c + d*x)^2),x)

[Out]

int(1/((a + b*(F^(g*(e + f*x)))^n)^3*(c + d*x)^2), x)

________________________________________________________________________________________